LMFDB - Embedded newform 1620.2.i.l.1081.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(541,1620)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1620.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.9357651274$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1620.1081
Dual form			1620.2.i.l.541.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{7} +O(q^{10})$	$q+(0.500000 - 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{7} +(1.50000 + 2.59808i) q^{11} +(2.00000 - 3.46410i) q^{13} +5.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-4.50000 - 7.79423i) q^{29} +(-2.50000 + 4.33013i) q^{31} +4.00000 q^{35} +2.00000 q^{37} +(-4.50000 + 7.79423i) q^{41} +(5.00000 + 8.66025i) q^{43} +(-3.00000 - 5.19615i) q^{47} +(-4.50000 + 7.79423i) q^{49} +12.0000 q^{53} +3.00000 q^{55} +(4.50000 - 7.79423i) q^{59} +(5.00000 + 8.66025i) q^{61} +(-2.00000 - 3.46410i) q^{65} +(-1.00000 + 1.73205i) q^{67} -3.00000 q^{71} -4.00000 q^{73} +(-6.00000 + 10.3923i) q^{77} +(2.00000 + 3.46410i) q^{79} +(3.00000 + 5.19615i) q^{83} +9.00000 q^{89} +16.0000 q^{91} +(2.50000 - 4.33013i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q + q^5 + 4 * q^7	$ 2 q + q^{5} + 4 q^{7} + 3 q^{11} + 4 q^{13} + 10 q^{19} - 6 q^{23} - q^{25} - 9 q^{29} - 5 q^{31} + 8 q^{35} + 4 q^{37} - 9 q^{41} + 10 q^{43} - 6 q^{47} - 9 q^{49} + 24 q^{53} + 6 q^{55} + 9 q^{59} + 10 q^{61} - 4 q^{65} - 2 q^{67} - 6 q^{71} - 8 q^{73} - 12 q^{77} + 4 q^{79} + 6 q^{83} + 18 q^{89} + 32 q^{91} + 5 q^{95} - 2 q^{97}+O(q^{100}) $ 2 * q + q^5 + 4 * q^7 + 3 * q^11 + 4 * q^13 + 10 * q^19 - 6 * q^23 - q^25 - 9 * q^29 - 5 * q^31 + 8 * q^35 + 4 * q^37 - 9 * q^41 + 10 * q^43 - 6 * q^47 - 9 * q^49 + 24 * q^53 + 6 * q^55 + 9 * q^59 + 10 * q^61 - 4 * q^65 - 2 * q^67 - 6 * q^71 - 8 * q^73 - 12 * q^77 + 4 * q^79 + 6 * q^83 + 18 * q^89 + 32 * q^91 + 5 * q^95 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	2.00000	+	3.46410i	0.755929	+	1.30931i	0.944911	+	0.327327i	$0.106148\pi$
$7$	2.00000	+	3.46410i	0.755929	+	1.30931i	−0.188982	+	0.981981i	$0.560519\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.50000	+	2.59808i	0.452267	+	0.783349i	0.998526	−	0.0542666i	$-0.0172821\pi$
$11$	1.50000	+	2.59808i	0.452267	+	0.783349i	−0.546259	+	0.837616i	$0.683949\pi$
$12$		0			0
$13$	2.00000	−	3.46410i	0.554700	−	0.960769i	−0.443227	−	0.896410i	$-0.646166\pi$
$13$	2.00000	−	3.46410i	0.554700	−	0.960769i	0.997927	−	0.0643593i	$-0.0205004\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	5.00000			1.14708			0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000			1.14708			0.573539	+	0.819178i	$0.305570\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.50000	−	7.79423i	−0.835629	−	1.44735i	−0.893517	−	0.449029i	$-0.851770\pi$
$29$	−4.50000	−	7.79423i	−0.835629	−	1.44735i	0.0578882	−	0.998323i	$-0.481563\pi$
$30$		0			0
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	−0.998322	−	0.0579057i	$-0.981558\pi$
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	0.549309	+	0.835619i	$0.314891\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.00000			0.676123
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.50000	+	7.79423i	−0.702782	+	1.21725i	0.264704	+	0.964330i	$0.414726\pi$
$41$	−4.50000	+	7.79423i	−0.702782	+	1.21725i	−0.967486	+	0.252924i	$0.918608\pi$
$42$		0			0
$43$	5.00000	+	8.66025i	0.762493	+	1.32068i	0.941562	+	0.336840i	$0.109358\pi$
$43$	5.00000	+	8.66025i	0.762493	+	1.32068i	−0.179069	+	0.983836i	$0.557309\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	0.559908	−	0.828554i	$-0.310836\pi$
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	−0.997503	+	0.0706177i	$0.977503\pi$
$48$		0			0
$49$	−4.50000	+	7.79423i	−0.642857	+	1.11346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.0000			1.64833			0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000			1.64833			0.824163	+	0.566352i	$0.191646\pi$
$54$		0			0
$55$	3.00000			0.404520
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.50000	−	7.79423i	0.585850	−	1.01472i	−0.408919	−	0.912571i	$-0.634094\pi$
$59$	4.50000	−	7.79423i	0.585850	−	1.01472i	0.994769	−	0.102151i	$-0.0325726\pi$
$60$		0			0
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	0.985391	+	0.170305i	$0.0544754\pi$
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	−0.345207	+	0.938527i	$0.612191\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000	−	3.46410i	−0.248069	−	0.429669i
$66$		0			0
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	−0.920623	−	0.390453i	$-0.872318\pi$
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	0.798454	+	0.602056i	$0.205652\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.00000			−0.356034			−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000			−0.356034			−0.178017	+	0.984027i	$0.556968\pi$
$72$		0			0
$73$	−4.00000			−0.468165			−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000			−0.468165			−0.234082	+	0.972217i	$0.575209\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.00000	+	10.3923i	−0.683763	+	1.18431i
$78$		0			0
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.00000	+	5.19615i	0.329293	+	0.570352i	0.982372	−	0.186938i	$-0.0598564\pi$
$83$	3.00000	+	5.19615i	0.329293	+	0.570352i	−0.653079	+	0.757290i	$0.726523\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.00000			0.953998			0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000			0.953998			0.476999	+	0.878904i	$0.341725\pi$
$90$		0			0
$91$	16.0000			1.67726
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.50000	−	4.33013i	0.256495	−	0.444262i
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.50000	−	12.9904i	−0.746278	−	1.29259i	−0.949595	−	0.313478i	$-0.898506\pi$
$101$	−7.50000	−	12.9904i	−0.746278	−	1.29259i	0.203317	−	0.979113i	$-0.434828\pi$
$102$		0			0
$103$	5.00000	−	8.66025i	0.492665	−	0.853320i	−0.507300	−	0.861770i	$-0.669356\pi$
$103$	5.00000	−	8.66025i	0.492665	−	0.853320i	0.999964	+	0.00844953i	$0.00268960\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.00000			−0.580042			−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000			−0.580042			−0.290021	+	0.957020i	$0.593662\pi$
$108$		0			0
$109$	11.0000			1.05361			0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000			1.05361			0.526804	+	0.849987i	$0.323390\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.00000	−	10.3923i	0.564433	−	0.977626i	−0.432670	−	0.901553i	$-0.642428\pi$
$113$	6.00000	−	10.3923i	0.564433	−	0.977626i	0.997102	−	0.0760733i	$-0.0242383\pi$
$114$		0			0
$115$	3.00000	+	5.19615i	0.279751	+	0.484544i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.00000	−	1.73205i	0.0909091	−	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	20.0000			1.77471			0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000			1.77471			0.887357	+	0.461084i	$0.152539\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.50000	+	12.9904i	−0.655278	+	1.13497i	0.326546	+	0.945181i	$0.394115\pi$
$131$	−7.50000	+	12.9904i	−0.655278	+	1.13497i	−0.981824	+	0.189794i	$0.939218\pi$
$132$		0			0
$133$	10.0000	+	17.3205i	0.867110	+	1.50188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	0.999893	+	0.0146279i	$0.00465636\pi$
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	−0.487278	+	0.873247i	$0.662010\pi$
$138$		0			0
$139$	6.50000	−	11.2583i	0.551323	−	0.954919i	−0.446857	−	0.894606i	$-0.647457\pi$
$139$	6.50000	−	11.2583i	0.551323	−	0.954919i	0.998179	−	0.0603135i	$-0.0192101\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	12.0000			1.00349
$144$		0			0
$145$	−9.00000			−0.747409
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.00000	+	15.5885i	−0.737309	+	1.27706i	0.216394	+	0.976306i	$0.430570\pi$
$149$	−9.00000	+	15.5885i	−0.737309	+	1.27706i	−0.953703	+	0.300750i	$0.902763\pi$
$150$		0			0
$151$	−8.50000	−	14.7224i	−0.691720	−	1.19809i	−0.971274	−	0.237964i	$-0.923520\pi$
$151$	−8.50000	−	14.7224i	−0.691720	−	1.19809i	0.279554	−	0.960130i	$-0.409814\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.50000	+	4.33013i	0.200805	+	0.347804i
$156$		0			0
$157$	−7.00000	+	12.1244i	−0.558661	+	0.967629i	0.438948	+	0.898513i	$0.355351\pi$
$157$	−7.00000	+	12.1244i	−0.558661	+	0.967629i	−0.997609	+	0.0691164i	$0.977982\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−24.0000			−1.89146
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	−1.50000	−	2.59808i	−0.115385	−	0.199852i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	0.729155	−	0.684349i	$-0.239913\pi$
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	−0.957241	+	0.289292i	$0.906580\pi$
$174$		0			0
$175$	2.00000	−	3.46410i	0.151186	−	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.00000			−0.672692			−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000			−0.672692			−0.336346	+	0.941739i	$0.609191\pi$
$180$		0			0
$181$	−13.0000			−0.966282			−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000			−0.966282			−0.483141	+	0.875542i	$0.660504\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.00000	−	1.73205i	0.0735215	−	0.127343i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.50000	+	2.59808i	0.108536	+	0.187990i	0.915177	−	0.403051i	$-0.132050\pi$
$191$	1.50000	+	2.59808i	0.108536	+	0.187990i	−0.806641	+	0.591041i	$0.798717\pi$
$192$		0			0
$193$	−7.00000	+	12.1244i	−0.503871	+	0.872730i	0.496119	+	0.868255i	$0.334758\pi$
$193$	−7.00000	+	12.1244i	−0.503871	+	0.872730i	−0.999990	+	0.00447566i	$0.998575\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	18.0000	−	31.1769i	1.26335	−	2.18819i
$204$		0			0
$205$	4.50000	+	7.79423i	0.314294	+	0.544373i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7.50000	+	12.9904i	0.518786	+	0.898563i
$210$		0			0
$211$	−2.50000	+	4.33013i	−0.172107	+	0.298098i	−0.939156	−	0.343490i	$-0.888391\pi$
$211$	−2.50000	+	4.33013i	−0.172107	+	0.298098i	0.767049	+	0.641588i	$0.221724\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	10.0000			0.681994
$216$		0			0
$217$	−20.0000			−1.35769
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4.00000	−	6.92820i	−0.267860	−	0.463947i	0.700449	−	0.713702i	$-0.252983\pi$
$223$	−4.00000	−	6.92820i	−0.267860	−	0.463947i	−0.968309	+	0.249756i	$0.919650\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.0000	−	20.7846i	−0.796468	−	1.37952i	−0.921903	−	0.387421i	$-0.873366\pi$
$227$	−12.0000	−	20.7846i	−0.796468	−	1.37952i	0.125435	−	0.992102i	$-0.459967\pi$
$228$		0			0
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	−0.999088	−	0.0426906i	$-0.986407\pi$
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	0.536515	+	0.843891i	$0.319740\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	30.0000			1.96537			0.982683	−	0.185296i	$-0.0593245\pi$
$233$	30.0000			1.96537			0.982683	+	0.185296i	$0.0593245\pi$
$234$		0			0
$235$	−6.00000			−0.391397
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.0000	+	20.7846i	−0.776215	+	1.34444i	0.157893	+	0.987456i	$0.449530\pi$
$239$	−12.0000	+	20.7846i	−0.776215	+	1.34444i	−0.934109	+	0.356988i	$0.883804\pi$
$240$		0			0
$241$	−14.5000	−	25.1147i	−0.934027	−	1.61778i	−0.776360	−	0.630290i	$-0.782936\pi$
$241$	−14.5000	−	25.1147i	−0.934027	−	1.61778i	−0.157667	−	0.987492i	$-0.550397\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.50000	+	7.79423i	0.287494	+	0.497955i
$246$		0			0
$247$	10.0000	−	17.3205i	0.636285	−	1.10208i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$	−18.0000			−1.13165
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.00000	+	5.19615i	−0.187135	+	0.324127i	−0.944294	−	0.329104i	$-0.893253\pi$
$257$	−3.00000	+	5.19615i	−0.187135	+	0.324127i	0.757159	+	0.653231i	$0.226587\pi$
$258$		0			0
$259$	4.00000	+	6.92820i	0.248548	+	0.430498i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.00000	+	5.19615i	0.184988	+	0.320408i	0.943572	−	0.331166i	$-0.107442\pi$
$263$	3.00000	+	5.19615i	0.184988	+	0.320408i	−0.758585	+	0.651575i	$0.774109\pi$
$264$		0			0
$265$	6.00000	−	10.3923i	0.368577	−	0.638394i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−21.0000			−1.28039			−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000			−1.28039			−0.640196	+	0.768211i	$0.721147\pi$
$270$		0			0
$271$	32.0000			1.94386			0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000			1.94386			0.971931	+	0.235267i	$0.0755965\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.50000	−	2.59808i	0.0904534	−	0.156670i
$276$		0			0
$277$	−13.0000	−	22.5167i	−0.781094	−	1.35290i	−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−13.0000	−	22.5167i	−0.781094	−	1.35290i	0.150210	−	0.988654i	$-0.452005\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	−0.999069	−	0.0431402i	$-0.986264\pi$
$281$	−9.00000	−	15.5885i	−0.536895	−	0.929929i	0.462174	−	0.886789i	$-0.347070\pi$
$282$		0			0
$283$	11.0000	−	19.0526i	0.653882	−	1.13256i	−0.328291	−	0.944577i	$-0.606473\pi$
$283$	11.0000	−	19.0526i	0.653882	−	1.13256i	0.982173	−	0.187980i	$-0.0601941\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−36.0000			−2.12501
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	15.0000	−	25.9808i	0.876309	−	1.51781i	0.0209480	−	0.999781i	$-0.493332\pi$
$293$	15.0000	−	25.9808i	0.876309	−	1.51781i	0.855361	−	0.518032i	$-0.173335\pi$
$294$		0			0
$295$	−4.50000	−	7.79423i	−0.262000	−	0.453798i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0000	+	20.7846i	0.693978	+	1.20201i
$300$		0			0
$301$	−20.0000	+	34.6410i	−1.15278	+	1.99667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	10.0000			0.572598
$306$		0			0
$307$	2.00000			0.114146			0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000			0.114146			0.0570730	+	0.998370i	$0.481823\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.50000	+	2.59808i	−0.0850572	+	0.147323i	−0.905416	−	0.424526i	$-0.860441\pi$
$311$	−1.50000	+	2.59808i	−0.0850572	+	0.147323i	0.820358	+	0.571850i	$0.193774\pi$
$312$		0			0
$313$	−10.0000	−	17.3205i	−0.565233	−	0.979013i	−0.997028	−	0.0770410i	$-0.975453\pi$
$313$	−10.0000	−	17.3205i	−0.565233	−	0.979013i	0.431795	−	0.901972i	$-0.357881\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	0.999980	+	0.00635137i	$0.00202172\pi$
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	−0.494489	+	0.869184i	$0.664645\pi$
$318$		0			0
$319$	13.5000	−	23.3827i	0.755855	−	1.30918i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−4.00000			−0.221880
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12.0000	−	20.7846i	0.661581	−	1.14589i
$330$		0			0
$331$	−2.50000	−	4.33013i	−0.137412	−	0.238005i	0.789104	−	0.614260i	$-0.210545\pi$
$331$	−2.50000	−	4.33013i	−0.137412	−	0.238005i	−0.926516	+	0.376254i	$0.877212\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.00000	+	1.73205i	0.0546358	+	0.0946320i
$336$		0			0
$337$	−4.00000	+	6.92820i	−0.217894	+	0.377403i	−0.954164	−	0.299285i	$-0.903252\pi$
$337$	−4.00000	+	6.92820i	−0.217894	+	0.377403i	0.736270	+	0.676688i	$0.236585\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−15.0000			−0.812296
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.0000	−	20.7846i	0.644194	−	1.11578i	−0.340293	−	0.940319i	$-0.610526\pi$
$347$	12.0000	−	20.7846i	0.644194	−	1.11578i	0.984487	−	0.175457i	$-0.0561403\pi$
$348$		0			0
$349$	−8.50000	−	14.7224i	−0.454995	−	0.788074i	0.543693	−	0.839284i	$-0.317025\pi$
$349$	−8.50000	−	14.7224i	−0.454995	−	0.788074i	−0.998688	+	0.0512103i	$0.983692\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.00000	−	5.19615i	−0.159674	−	0.276563i	0.775077	−	0.631867i	$-0.217711\pi$
$353$	−3.00000	−	5.19615i	−0.159674	−	0.276563i	−0.934751	+	0.355303i	$0.884378\pi$
$354$		0			0
$355$	−1.50000	+	2.59808i	−0.0796117	+	0.137892i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.0000			−0.791670			−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000			−0.791670			−0.395835	+	0.918322i	$0.629545\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.00000	+	3.46410i	−0.104685	+	0.181319i
$366$		0			0
$367$	−7.00000	−	12.1244i	−0.365397	−	0.632886i	0.623443	−	0.781869i	$-0.285733\pi$
$367$	−7.00000	−	12.1244i	−0.365397	−	0.632886i	−0.988840	+	0.148983i	$0.952400\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	24.0000	+	41.5692i	1.24602	+	2.15817i
$372$		0			0
$373$	14.0000	−	24.2487i	0.724893	−	1.25555i	−0.234126	−	0.972206i	$-0.575223\pi$
$373$	14.0000	−	24.2487i	0.724893	−	1.25555i	0.959018	−	0.283344i	$-0.0914439\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−36.0000			−1.85409
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	18.0000	−	31.1769i	0.919757	−	1.59307i	0.119974	−	0.992777i	$-0.461719\pi$
$383$	18.0000	−	31.1769i	0.919757	−	1.59307i	0.799783	−	0.600289i	$-0.204948\pi$
$384$		0			0
$385$	6.00000	+	10.3923i	0.305788	+	0.529641i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.00000	−	15.5885i	−0.456318	−	0.790366i	0.542445	−	0.840091i	$-0.317499\pi$
$389$	−9.00000	−	15.5885i	−0.456318	−	0.790366i	−0.998763	+	0.0497253i	$0.984165\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.00000			0.201262
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	−0.998350	−	0.0574304i	$-0.981709\pi$
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	0.548911	+	0.835881i	$0.315043\pi$
$402$		0			0
$403$	10.0000	+	17.3205i	0.498135	+	0.862796i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.00000	+	5.19615i	0.148704	+	0.257564i
$408$		0			0
$409$	5.00000	−	8.66025i	0.247234	−	0.428222i	−0.715523	−	0.698589i	$-0.753812\pi$
$409$	5.00000	−	8.66025i	0.247234	−	0.428222i	0.962757	+	0.270367i	$0.0871450\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	36.0000			1.77144
$414$		0			0
$415$	6.00000			0.294528
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$	−8.50000	−	14.7224i	−0.414265	−	0.717527i	0.581086	−	0.813842i	$-0.302628\pi$
$421$	−8.50000	−	14.7224i	−0.414265	−	0.717527i	−0.995351	+	0.0963145i	$0.969295\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−20.0000	+	34.6410i	−0.967868	+	1.67640i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−33.0000			−1.58955			−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000			−1.58955			−0.794777	+	0.606902i	$0.792412\pi$
$432$		0			0
$433$	−34.0000			−1.63394			−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000			−1.63394			−0.816968	+	0.576683i	$0.804347\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−15.0000	+	25.9808i	−0.717547	+	1.24283i
$438$		0			0
$439$	−8.50000	−	14.7224i	−0.405683	−	0.702663i	0.588718	−	0.808339i	$-0.299633\pi$
$439$	−8.50000	−	14.7224i	−0.405683	−	0.702663i	−0.994401	+	0.105675i	$0.966300\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.00000	+	5.19615i	0.142534	+	0.246877i	0.928450	−	0.371457i	$-0.121142\pi$
$443$	3.00000	+	5.19615i	0.142534	+	0.246877i	−0.785916	+	0.618333i	$0.787808\pi$
$444$		0			0
$445$	4.50000	−	7.79423i	0.213320	−	0.369482i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	39.0000			1.84052			0.920262	−	0.391303i	$-0.127976\pi$
$449$	39.0000			1.84052			0.920262	+	0.391303i	$0.127976\pi$
$450$		0			0
$451$	−27.0000			−1.27138
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.00000	−	13.8564i	0.375046	−	0.649598i
$456$		0			0
$457$	5.00000	+	8.66025i	0.233890	+	0.405110i	0.958950	−	0.283577i	$-0.0915211\pi$
$457$	5.00000	+	8.66025i	0.233890	+	0.405110i	−0.725059	+	0.688686i	$0.758188\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.50000	−	2.59808i	−0.0698620	−	0.121004i	0.828978	−	0.559281i	$-0.188923\pi$
$461$	−1.50000	−	2.59808i	−0.0698620	−	0.121004i	−0.898840	+	0.438276i	$0.855589\pi$
$462$		0			0
$463$	17.0000	−	29.4449i	0.790057	−	1.36842i	−0.135874	−	0.990726i	$-0.543384\pi$
$463$	17.0000	−	29.4449i	0.790057	−	1.36842i	0.925931	−	0.377693i	$-0.123282\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−15.0000	+	25.9808i	−0.689701	+	1.19460i
$474$		0			0
$475$	−2.50000	−	4.33013i	−0.114708	−	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.50000	+	2.59808i	0.0685367	+	0.118709i	0.898257	−	0.439470i	$-0.144834\pi$
$479$	1.50000	+	2.59808i	0.0685367	+	0.118709i	−0.829721	+	0.558179i	$0.811500\pi$
$480$		0			0
$481$	4.00000	−	6.92820i	0.182384	−	0.315899i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.00000			−0.0908153
$486$		0			0
$487$	−16.0000			−0.725029			−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000			−0.725029			−0.362515	+	0.931978i	$0.618082\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.50000	+	2.59808i	−0.0676941	+	0.117250i	−0.897886	−	0.440228i	$-0.854898\pi$
$491$	−1.50000	+	2.59808i	−0.0676941	+	0.117250i	0.830192	+	0.557478i	$0.188231\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.00000	−	10.3923i	−0.269137	−	0.466159i
$498$		0			0
$499$	−20.5000	+	35.5070i	−0.917706	+	1.58951i	−0.114816	+	0.993387i	$0.536628\pi$
$499$	−20.5000	+	35.5070i	−0.917706	+	1.58951i	−0.802890	+	0.596127i	$0.796706\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	36.0000			1.60516			0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000			1.60516			0.802580	+	0.596544i	$0.203460\pi$
$504$		0			0
$505$	−15.0000			−0.667491
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.00000	−	15.5885i	0.398918	−	0.690946i	−0.594675	−	0.803966i	$-0.702719\pi$
$509$	9.00000	−	15.5885i	0.398918	−	0.690946i	0.993593	+	0.113020i	$0.0360525\pi$
$510$		0			0
$511$	−8.00000	−	13.8564i	−0.353899	−	0.612971i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.00000	−	8.66025i	−0.220326	−	0.381616i
$516$		0			0
$517$	9.00000	−	15.5885i	0.395820	−	0.685580i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		0			0
$523$	−16.0000			−0.699631			−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000			−0.699631			−0.349816	+	0.936819i	$0.613756\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	18.0000	+	31.1769i	0.779667	+	1.35042i
$534$		0			0
$535$	−3.00000	+	5.19615i	−0.129701	+	0.224649i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−27.0000			−1.16297
$540$		0			0
$541$	−1.00000			−0.0429934			−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000			−0.0429934			−0.0214967	+	0.999769i	$0.506843\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.50000	−	9.52628i	0.235594	−	0.408061i
$546$		0			0
$547$	−10.0000	−	17.3205i	−0.427569	−	0.740571i	0.569087	−	0.822277i	$-0.307297\pi$
$547$	−10.0000	−	17.3205i	−0.427569	−	0.740571i	−0.996657	+	0.0817056i	$0.973963\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−22.5000	−	38.9711i	−0.958532	−	1.66023i
$552$		0			0
$553$	−8.00000	+	13.8564i	−0.340195	+	0.589234i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	24.0000			1.01691			0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000			1.01691			0.508456	+	0.861088i	$0.330216\pi$
$558$		0			0
$559$	40.0000			1.69182
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9.00000	+	15.5885i	−0.379305	+	0.656975i	−0.990961	−	0.134148i	$-0.957170\pi$
$563$	−9.00000	+	15.5885i	−0.379305	+	0.656975i	0.611656	+	0.791123i	$0.290503\pi$
$564$		0			0
$565$	−6.00000	−	10.3923i	−0.252422	−	0.437208i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13.5000	−	23.3827i	−0.565949	−	0.980253i	−0.996961	−	0.0779066i	$-0.975176\pi$
$569$	−13.5000	−	23.3827i	−0.565949	−	0.980253i	0.431011	−	0.902347i	$-0.358157\pi$
$570$		0			0
$571$	−5.50000	+	9.52628i	−0.230168	+	0.398662i	−0.957857	−	0.287244i	$-0.907261\pi$
$571$	−5.50000	+	9.52628i	−0.230168	+	0.398662i	0.727690	+	0.685907i	$0.240594\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.00000			0.250217
$576$		0			0
$577$	20.0000			0.832611			0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000			0.832611			0.416305	+	0.909225i	$0.363325\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.0000	+	20.7846i	−0.497844	+	0.862291i
$582$		0			0
$583$	18.0000	+	31.1769i	0.745484	+	1.29122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9.00000	+	15.5885i	0.371470	+	0.643404i	0.989792	−	0.142520i	$-0.0455206\pi$
$587$	9.00000	+	15.5885i	0.371470	+	0.643404i	−0.618322	+	0.785925i	$0.712187\pi$
$588$		0			0
$589$	−12.5000	+	21.6506i	−0.515054	+	0.892099i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−42.0000			−1.72473			−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000			−1.72473			−0.862367	+	0.506284i	$0.831019\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.50000	+	2.59808i	−0.0612883	+	0.106155i	−0.895042	−	0.445983i	$-0.852854\pi$
$599$	−1.50000	+	2.59808i	−0.0612883	+	0.106155i	0.833753	+	0.552137i	$0.186188\pi$
$600$		0			0
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	0.992114	−	0.125336i	$-0.0400009\pi$
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	−0.604601	+	0.796528i	$0.706668\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.00000	−	1.73205i	−0.0406558	−	0.0704179i
$606$		0			0
$607$	11.0000	−	19.0526i	0.446476	−	0.773320i	−0.551678	−	0.834058i	$-0.686012\pi$
$607$	11.0000	−	19.0526i	0.446476	−	0.773320i	0.998154	+	0.0607380i	$0.0193454\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−24.0000			−0.970936
$612$		0			0
$613$	−4.00000			−0.161558			−0.0807792	−	0.996732i	$-0.525741\pi$
$613$	−4.00000			−0.161558			−0.0807792	+	0.996732i	$0.525741\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−24.0000	+	41.5692i	−0.966204	+	1.67351i	−0.259858	+	0.965647i	$0.583676\pi$
$617$	−24.0000	+	41.5692i	−0.966204	+	1.67351i	−0.706346	+	0.707867i	$0.749658\pi$
$618$		0			0
$619$	14.0000	+	24.2487i	0.562708	+	0.974638i	0.997259	+	0.0739910i	$0.0235736\pi$
$619$	14.0000	+	24.2487i	0.562708	+	0.974638i	−0.434551	+	0.900647i	$0.643093\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18.0000	+	31.1769i	0.721155	+	1.24908i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	5.00000			0.199047			0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000			0.199047			0.0995234	+	0.995035i	$0.468268\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10.0000	−	17.3205i	0.396838	−	0.687343i
$636$		0			0
$637$	18.0000	+	31.1769i	0.713186	+	1.23527i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.50000	+	2.59808i	0.0592464	+	0.102618i	0.894127	−	0.447813i	$-0.147797\pi$
$641$	1.50000	+	2.59808i	0.0592464	+	0.102618i	−0.834881	+	0.550431i	$0.814464\pi$
$642$		0			0
$643$	−7.00000	+	12.1244i	−0.276053	+	0.478138i	−0.970400	−	0.241502i	$-0.922360\pi$
$643$	−7.00000	+	12.1244i	−0.276053	+	0.478138i	0.694347	+	0.719640i	$0.255693\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	30.0000			1.17942			0.589711	−	0.807614i	$-0.299242\pi$
$647$	30.0000			1.17942			0.589711	+	0.807614i	$0.299242\pi$
$648$		0			0
$649$	27.0000			1.05984
$650$		0			0
$651$		0			0
$652$		0			0
$653$	24.0000	−	41.5692i	0.939193	−	1.62673i	0.172211	−	0.985060i	$-0.444909\pi$
$653$	24.0000	−	41.5692i	0.939193	−	1.62673i	0.766982	−	0.641669i	$-0.221758\pi$
$654$		0			0
$655$	7.50000	+	12.9904i	0.293049	+	0.507576i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	0.725175	−	0.688565i	$-0.241759\pi$
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	−0.958902	+	0.283738i	$0.908425\pi$
$660$		0			0
$661$	9.50000	−	16.4545i	0.369507	−	0.640005i	−0.619981	−	0.784617i	$-0.712860\pi$
$661$	9.50000	−	16.4545i	0.369507	−	0.640005i	0.989489	+	0.144611i	$0.0461932\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	20.0000			0.775567
$666$		0			0
$667$	54.0000			2.09089
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−15.0000	+	25.9808i	−0.579069	+	1.00298i
$672$		0			0
$673$	11.0000	+	19.0526i	0.424019	+	0.734422i	0.996328	−	0.0856156i	$-0.0272857\pi$
$673$	11.0000	+	19.0526i	0.424019	+	0.734422i	−0.572309	+	0.820038i	$0.693952\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.00000	+	5.19615i	0.115299	+	0.199704i	0.917899	−	0.396813i	$-0.129884\pi$
$677$	3.00000	+	5.19615i	0.115299	+	0.199704i	−0.802600	+	0.596518i	$0.796551\pi$
$678$		0			0
$679$	4.00000	−	6.92820i	0.153506	−	0.265880i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	24.0000			0.918334			0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000			0.918334			0.459167	+	0.888350i	$0.348148\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$		0			0
$687$		0			0
$688$		0			0
$689$	24.0000	−	41.5692i	0.914327	−	1.58366i
$690$		0			0
$691$	14.0000	+	24.2487i	0.532585	+	0.922464i	0.999276	+	0.0380440i	$0.0121127\pi$
$691$	14.0000	+	24.2487i	0.532585	+	0.922464i	−0.466691	+	0.884420i	$0.654554\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.50000	−	11.2583i	−0.246559	−	0.427053i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	9.00000			0.339925			0.169963	−	0.985451i	$-0.445635\pi$
$701$	9.00000			0.339925			0.169963	+	0.985451i	$0.445635\pi$
$702$		0			0
$703$	10.0000			0.377157
$704$		0			0
$705$		0			0
$706$		0			0
$707$	30.0000	−	51.9615i	1.12827	−	1.95421i
$708$		0			0
$709$	−19.0000	−	32.9090i	−0.713560	−	1.23592i	−0.963512	−	0.267664i	$-0.913748\pi$
$709$	−19.0000	−	32.9090i	−0.713560	−	1.23592i	0.249952	−	0.968258i	$-0.419585\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−15.0000	−	25.9808i	−0.561754	−	0.972987i
$714$		0			0
$715$	6.00000	−	10.3923i	0.224387	−	0.388650i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	15.0000			0.559406			0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000			0.559406			0.279703	+	0.960087i	$0.409764\pi$
$720$		0			0
$721$	40.0000			1.48968
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.50000	+	7.79423i	−0.167126	+	0.289470i
$726$		0			0
$727$	−22.0000	−	38.1051i	−0.815935	−	1.41324i	−0.908655	−	0.417548i	$-0.862889\pi$
$727$	−22.0000	−	38.1051i	−0.815935	−	1.41324i	0.0927199	−	0.995692i	$-0.470444\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	11.0000	−	19.0526i	0.406294	−	0.703722i	−0.588177	−	0.808732i	$-0.700154\pi$
$733$	11.0000	−	19.0526i	0.406294	−	0.703722i	0.994471	+	0.105010i	$0.0334875\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.00000			−0.221013
$738$		0			0
$739$	11.0000			0.404642			0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000			0.404642			0.202321	+	0.979319i	$0.435152\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−24.0000	+	41.5692i	−0.880475	+	1.52503i	−0.0296605	+	0.999560i	$0.509443\pi$
$743$	−24.0000	+	41.5692i	−0.880475	+	1.52503i	−0.850814	+	0.525467i	$0.823891\pi$
$744$		0			0
$745$	9.00000	+	15.5885i	0.329734	+	0.571117i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12.0000	−	20.7846i	−0.438470	−	0.759453i
$750$		0			0
$751$	20.0000	−	34.6410i	0.729810	−	1.26407i	−0.227153	−	0.973859i	$-0.572942\pi$
$751$	20.0000	−	34.6410i	0.729810	−	1.26407i	0.956963	−	0.290209i	$-0.0937250\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−17.0000			−0.618693
$756$		0			0
$757$	−52.0000			−1.88997			−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−52.0000			−1.88997			−0.944986	+	0.327111i	$0.893925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.50000	+	2.59808i	−0.0543750	+	0.0941802i	−0.891932	−	0.452170i	$-0.850650\pi$
$761$	−1.50000	+	2.59808i	−0.0543750	+	0.0941802i	0.837557	+	0.546350i	$0.183983\pi$
$762$		0			0
$763$	22.0000	+	38.1051i	0.796453	+	1.37950i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.0000	−	31.1769i	−0.649942	−	1.12573i
$768$		0			0
$769$	−2.50000	+	4.33013i	−0.0901523	+	0.156148i	−0.907575	−	0.419890i	$-0.862069\pi$
$769$	−2.50000	+	4.33013i	−0.0901523	+	0.156148i	0.817423	+	0.576038i	$0.195402\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.00000			−0.215805			−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000			−0.215805			−0.107903	+	0.994161i	$0.534413\pi$
$774$		0			0
$775$	5.00000			0.179605
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−22.5000	+	38.9711i	−0.806146	+	1.39629i
$780$		0			0
$781$	−4.50000	−	7.79423i	−0.161023	−	0.278899i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.00000	+	12.1244i	0.249841	+	0.432737i
$786$		0			0
$787$	−16.0000	+	27.7128i	−0.570338	+	0.987855i	0.426193	+	0.904632i	$0.359855\pi$
$787$	−16.0000	+	27.7128i	−0.570338	+	0.987855i	−0.996531	+	0.0832226i	$0.973479\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	48.0000			1.70668
$792$		0			0
$793$	40.0000			1.42044
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−6.00000	+	10.3923i	−0.212531	+	0.368114i	−0.952506	−	0.304520i	$-0.901504\pi$
$797$	−6.00000	+	10.3923i	−0.212531	+	0.368114i	0.739975	+	0.672634i	$0.234837\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.00000	−	10.3923i	−0.211735	−	0.366736i
$804$		0			0
$805$	−12.0000	+	20.7846i	−0.422944	+	0.732561i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9.00000			0.316423			0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000			0.316423			0.158212	+	0.987405i	$0.449427\pi$
$810$		0			0
$811$	29.0000			1.01833			0.509164	−	0.860670i	$-0.329955\pi$
$811$	29.0000			1.01833			0.509164	+	0.860670i	$0.329955\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.00000	+	3.46410i	−0.0700569	+	0.121342i
$816$		0			0
$817$	25.0000	+	43.3013i	0.874639	+	1.51492i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7.50000	+	12.9904i	0.261752	+	0.453367i	0.966708	−	0.255884i	$-0.0823665\pi$
$821$	7.50000	+	12.9904i	0.261752	+	0.453367i	−0.704956	+	0.709251i	$0.749033\pi$
$822$		0			0
$823$	−7.00000	+	12.1244i	−0.244005	+	0.422628i	−0.961851	−	0.273573i	$-0.911795\pi$
$823$	−7.00000	+	12.1244i	−0.244005	+	0.422628i	0.717847	+	0.696201i	$0.245128\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−12.0000			−0.417281			−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000			−0.417281			−0.208640	+	0.977992i	$0.566904\pi$
$828$		0			0
$829$	−1.00000			−0.0347314			−0.0173657	−	0.999849i	$-0.505528\pi$
$829$	−1.00000			−0.0347314			−0.0173657	+	0.999849i	$0.505528\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	7.50000	+	12.9904i	0.258929	+	0.448478i	0.965955	−	0.258709i	$-0.0832972\pi$
$839$	7.50000	+	12.9904i	0.258929	+	0.448478i	−0.707026	+	0.707187i	$0.749964\pi$
$840$		0			0
$841$	−26.0000	+	45.0333i	−0.896552	+	1.55287i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.00000			−0.103203
$846$		0			0
$847$	8.00000			0.274883
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.00000	+	10.3923i	−0.205677	+	0.356244i
$852$		0			0
$853$	2.00000	+	3.46410i	0.0684787	+	0.118609i	0.898232	−	0.439522i	$-0.144852\pi$
$853$	2.00000	+	3.46410i	0.0684787	+	0.118609i	−0.829753	+	0.558131i	$0.811519\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	15.5000	−	26.8468i	0.528853	−	0.916001i	−0.470581	−	0.882357i	$-0.655956\pi$
$859$	15.5000	−	26.8468i	0.528853	−	0.916001i	0.999434	−	0.0336436i	$-0.0107111\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24.0000			0.816970			0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$		0			0
$865$	−6.00000			−0.204006
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6.00000	+	10.3923i	−0.203536	+	0.352535i
$870$		0			0
$871$	4.00000	+	6.92820i	0.135535	+	0.234753i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.00000	−	3.46410i	−0.0676123	−	0.117108i
$876$		0			0
$877$	8.00000	−	13.8564i	0.270141	−	0.467898i	−0.698757	−	0.715359i	$-0.746263\pi$
$877$	8.00000	−	13.8564i	0.270141	−	0.467898i	0.968898	+	0.247462i	$0.0795964\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	33.0000			1.11180			0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000			1.11180			0.555899	+	0.831250i	$0.312374\pi$
$882$		0			0
$883$	−40.0000			−1.34611			−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000			−1.34611			−0.673054	+	0.739594i	$0.735018\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−21.0000	+	36.3731i	−0.705111	+	1.22129i	0.261540	+	0.965193i	$0.415770\pi$
$887$	−21.0000	+	36.3731i	−0.705111	+	1.22129i	−0.966651	+	0.256096i	$0.917564\pi$
$888$		0			0
$889$	40.0000	+	69.2820i	1.34156	+	2.32364i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−15.0000	−	25.9808i	−0.501956	−	0.869413i
$894$		0			0
$895$	−4.50000	+	7.79423i	−0.150418	+	0.260532i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	45.0000			1.50083
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.50000	+	11.2583i	−0.216067	+	0.374240i
$906$		0			0
$907$	11.0000	+	19.0526i	0.365249	+	0.632630i	0.988816	−	0.149140i	$-0.0476505\pi$
$907$	11.0000	+	19.0526i	0.365249	+	0.632630i	−0.623567	+	0.781770i	$0.714317\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−25.5000	−	44.1673i	−0.844853	−	1.46333i	−0.885749	−	0.464164i	$-0.846355\pi$
$911$	−25.5000	−	44.1673i	−0.844853	−	1.46333i	0.0408964	−	0.999163i	$-0.486979\pi$
$912$		0			0
$913$	−9.00000	+	15.5885i	−0.297857	+	0.515903i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−60.0000			−1.98137
$918$		0			0
$919$	−25.0000			−0.824674			−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000			−0.824674			−0.412337	+	0.911031i	$0.635287\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−6.00000	+	10.3923i	−0.197492	+	0.342067i
$924$		0			0
$925$	−1.00000	−	1.73205i	−0.0328798	−	0.0569495i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−7.50000	−	12.9904i	−0.246067	−	0.426201i	0.716364	−	0.697727i	$-0.245805\pi$
$929$	−7.50000	−	12.9904i	−0.246067	−	0.426201i	−0.962431	+	0.271526i	$0.912472\pi$
$930$		0			0
$931$	−22.5000	+	38.9711i	−0.737408	+	1.27723i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	26.0000			0.849383			0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000			0.849383			0.424691	+	0.905338i	$0.360383\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.00000	−	5.19615i	0.0977972	−	0.169390i	−0.812975	−	0.582298i	$-0.802154\pi$
$941$	3.00000	−	5.19615i	0.0977972	−	0.169390i	0.910773	+	0.412908i	$0.135487\pi$
$942$		0			0
$943$	−27.0000	−	46.7654i	−0.879241	−	1.52289i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.00000	−	15.5885i	−0.292461	−	0.506557i	0.681930	−	0.731417i	$-0.261141\pi$
$947$	−9.00000	−	15.5885i	−0.292461	−	0.506557i	−0.974391	+	0.224860i	$0.927807\pi$
$948$		0			0
$949$	−8.00000	+	13.8564i	−0.259691	+	0.449798i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	24.0000			0.777436			0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000			0.777436			0.388718	+	0.921357i	$0.372918\pi$
$954$		0			0
$955$	3.00000			0.0970777
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−24.0000	+	41.5692i	−0.775000	+	1.34234i
$960$		0			0
$961$	3.00000	+	5.19615i	0.0967742	+	0.167618i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	7.00000	+	12.1244i	0.225338	+	0.390297i
$966$		0			0
$967$	−22.0000	+	38.1051i	−0.707472	+	1.22538i	0.258320	+	0.966060i	$0.416831\pi$
$967$	−22.0000	+	38.1051i	−0.707472	+	1.22538i	−0.965792	+	0.259318i	$0.916502\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.00000			−0.0962746			−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000			−0.0962746			−0.0481373	+	0.998841i	$0.515328\pi$
$972$		0			0
$973$	52.0000			1.66704
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6.00000	+	10.3923i	−0.191957	+	0.332479i	−0.945899	−	0.324462i	$-0.894817\pi$
$977$	−6.00000	+	10.3923i	−0.191957	+	0.332479i	0.753942	+	0.656941i	$0.228150\pi$
$978$		0			0
$979$	13.5000	+	23.3827i	0.431462	+	0.747314i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.00000	−	10.3923i	−0.191370	−	0.331463i	0.754334	−	0.656490i	$-0.227960\pi$
$983$	−6.00000	−	10.3923i	−0.191370	−	0.331463i	−0.945705	+	0.325027i	$0.894626\pi$
$984$		0			0
$985$	6.00000	−	10.3923i	0.191176	−	0.331126i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−60.0000			−1.90789
$990$		0			0
$991$	−43.0000			−1.36594			−0.682970	−	0.730446i	$-0.739312\pi$
$991$	−43.0000			−1.36594			−0.682970	+	0.730446i	$0.739312\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.00000	+	13.8564i	−0.253617	+	0.439278i
$996$		0			0
$997$	14.0000	+	24.2487i	0.443384	+	0.767964i	0.997938	−	0.0641836i	$-0.0204443\pi$
$997$	14.0000	+	24.2487i	0.443384	+	0.767964i	−0.554554	+	0.832148i	$0.687111\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.2.i.l.1081.1		2
3.2	odd	2		1620.2.i.e.1081.1		2
9.2	odd	6		1620.2.i.e.541.1		2
9.4	even	3		1620.2.a.a.1.1	✓	1
9.5	odd	6		1620.2.a.d.1.1	yes	1
9.7	even	3	inner	1620.2.i.l.541.1		2
36.23	even	6		6480.2.a.y.1.1		1
36.31	odd	6		6480.2.a.m.1.1		1
45.4	even	6		8100.2.a.m.1.1		1
45.13	odd	12		8100.2.d.b.649.2		2
45.14	odd	6		8100.2.a.n.1.1		1
45.22	odd	12		8100.2.d.b.649.1		2
45.23	even	12		8100.2.d.g.649.2		2
45.32	even	12		8100.2.d.g.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.2.a.a.1.1	✓	1	9.4	even	3
1620.2.a.d.1.1	yes	1	9.5	odd	6
1620.2.i.e.541.1		2	9.2	odd	6
1620.2.i.e.1081.1		2	3.2	odd	2
1620.2.i.l.541.1		2	9.7	even	3	inner
1620.2.i.l.1081.1		2	1.1	even	1	trivial
6480.2.a.m.1.1		1	36.31	odd	6
6480.2.a.y.1.1		1	36.23	even	6
8100.2.a.m.1.1		1	45.4	even	6
8100.2.a.n.1.1		1	45.14	odd	6
8100.2.d.b.649.1		2	45.22	odd	12
8100.2.d.b.649.2		2	45.13	odd	12
8100.2.d.g.649.1		2	45.32	even	12
8100.2.d.g.649.2		2	45.23	even	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{7} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{7} +(1.50000 + 2.59808i) q^{11} +(2.00000 - 3.46410i) q^{13} +5.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-4.50000 - 7.79423i) q^{29} +(-2.50000 + 4.33013i) q^{31} +4.00000 q^{35} +2.00000 q^{37} +(-4.50000 + 7.79423i) q^{41} +(5.00000 + 8.66025i) q^{43} +(-3.00000 - 5.19615i) q^{47} +(-4.50000 + 7.79423i) q^{49} +12.0000 q^{53} +3.00000 q^{55} +(4.50000 - 7.79423i) q^{59} +(5.00000 + 8.66025i) q^{61} +(-2.00000 - 3.46410i) q^{65} +(-1.00000 + 1.73205i) q^{67} -3.00000 q^{71} -4.00000 q^{73} +(-6.00000 + 10.3923i) q^{77} +(2.00000 + 3.46410i) q^{79} +(3.00000 + 5.19615i) q^{83} +9.00000 q^{89} +16.0000 q^{91} +(2.50000 - 4.33013i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q + q^5 + 4 * q^7	\( 2 q + q^{5} + 4 q^{7} + 3 q^{11} + 4 q^{13} + 10 q^{19} - 6 q^{23} - q^{25} - 9 q^{29} - 5 q^{31} + 8 q^{35} + 4 q^{37} - 9 q^{41} + 10 q^{43} - 6 q^{47} - 9 q^{49} + 24 q^{53} + 6 q^{55} + 9 q^{59} + 10 q^{61} - 4 q^{65} - 2 q^{67} - 6 q^{71} - 8 q^{73} - 12 q^{77} + 4 q^{79} + 6 q^{83} + 18 q^{89} + 32 q^{91} + 5 q^{95} - 2 q^{97}+O(q^{100}) \) 2 * q + q^5 + 4 * q^7 + 3 * q^11 + 4 * q^13 + 10 * q^19 - 6 * q^23 - q^25 - 9 * q^29 - 5 * q^31 + 8 * q^35 + 4 * q^37 - 9 * q^41 + 10 * q^43 - 6 * q^47 - 9 * q^49 + 24 * q^53 + 6 * q^55 + 9 * q^59 + 10 * q^61 - 4 * q^65 - 2 * q^67 - 6 * q^71 - 8 * q^73 - 12 * q^77 + 4 * q^79 + 6 * q^83 + 18 * q^89 + 32 * q^91 + 5 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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